Quiver Schur algebras for the linear quiver I

Jun Hu and Andrew Mathas

Abstract

We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras \(\mathcal{R}^\Lambda_n\) of type \(A\) when \(e=0\) (the linear quiver) or \(e\ge n\). We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When \(e=0\) we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category \(\mathcal O^\Lambda_n\) previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when \(e=0\) our graded Schur algebras are Koszul over field of characteristic zero. Finally, we give an LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when \(e=0\).

Keywords: Cyclotomic Hecke algebras, Schur algebras, quasi-hereditary and graded cellular algebras, Khovanov-Lauda-Rouquier algebras.

AMS Subject Classification: Primary 20C08; secondary 20C30, 05E10.